# Properties

 Label 525.4.a.a Level $525$ Weight $4$ Character orbit 525.a Self dual yes Analytic conductor $30.976$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,4,Mod(1,525)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("525.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$30.9760027530$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 5 q^{2} + 3 q^{3} + 17 q^{4} - 15 q^{6} - 7 q^{7} - 45 q^{8} + 9 q^{9}+O(q^{10})$$ q - 5 * q^2 + 3 * q^3 + 17 * q^4 - 15 * q^6 - 7 * q^7 - 45 * q^8 + 9 * q^9 $$q - 5 q^{2} + 3 q^{3} + 17 q^{4} - 15 q^{6} - 7 q^{7} - 45 q^{8} + 9 q^{9} + 12 q^{11} + 51 q^{12} - 30 q^{13} + 35 q^{14} + 89 q^{16} + 134 q^{17} - 45 q^{18} - 92 q^{19} - 21 q^{21} - 60 q^{22} - 112 q^{23} - 135 q^{24} + 150 q^{26} + 27 q^{27} - 119 q^{28} - 58 q^{29} - 224 q^{31} - 85 q^{32} + 36 q^{33} - 670 q^{34} + 153 q^{36} + 146 q^{37} + 460 q^{38} - 90 q^{39} + 18 q^{41} + 105 q^{42} - 340 q^{43} + 204 q^{44} + 560 q^{46} - 208 q^{47} + 267 q^{48} + 49 q^{49} + 402 q^{51} - 510 q^{52} + 754 q^{53} - 135 q^{54} + 315 q^{56} - 276 q^{57} + 290 q^{58} + 380 q^{59} + 718 q^{61} + 1120 q^{62} - 63 q^{63} - 287 q^{64} - 180 q^{66} - 412 q^{67} + 2278 q^{68} - 336 q^{69} - 960 q^{71} - 405 q^{72} - 1066 q^{73} - 730 q^{74} - 1564 q^{76} - 84 q^{77} + 450 q^{78} + 896 q^{79} + 81 q^{81} - 90 q^{82} - 436 q^{83} - 357 q^{84} + 1700 q^{86} - 174 q^{87} - 540 q^{88} - 1038 q^{89} + 210 q^{91} - 1904 q^{92} - 672 q^{93} + 1040 q^{94} - 255 q^{96} + 702 q^{97} - 245 q^{98} + 108 q^{99}+O(q^{100})$$ q - 5 * q^2 + 3 * q^3 + 17 * q^4 - 15 * q^6 - 7 * q^7 - 45 * q^8 + 9 * q^9 + 12 * q^11 + 51 * q^12 - 30 * q^13 + 35 * q^14 + 89 * q^16 + 134 * q^17 - 45 * q^18 - 92 * q^19 - 21 * q^21 - 60 * q^22 - 112 * q^23 - 135 * q^24 + 150 * q^26 + 27 * q^27 - 119 * q^28 - 58 * q^29 - 224 * q^31 - 85 * q^32 + 36 * q^33 - 670 * q^34 + 153 * q^36 + 146 * q^37 + 460 * q^38 - 90 * q^39 + 18 * q^41 + 105 * q^42 - 340 * q^43 + 204 * q^44 + 560 * q^46 - 208 * q^47 + 267 * q^48 + 49 * q^49 + 402 * q^51 - 510 * q^52 + 754 * q^53 - 135 * q^54 + 315 * q^56 - 276 * q^57 + 290 * q^58 + 380 * q^59 + 718 * q^61 + 1120 * q^62 - 63 * q^63 - 287 * q^64 - 180 * q^66 - 412 * q^67 + 2278 * q^68 - 336 * q^69 - 960 * q^71 - 405 * q^72 - 1066 * q^73 - 730 * q^74 - 1564 * q^76 - 84 * q^77 + 450 * q^78 + 896 * q^79 + 81 * q^81 - 90 * q^82 - 436 * q^83 - 357 * q^84 + 1700 * q^86 - 174 * q^87 - 540 * q^88 - 1038 * q^89 + 210 * q^91 - 1904 * q^92 - 672 * q^93 + 1040 * q^94 - 255 * q^96 + 702 * q^97 - 245 * q^98 + 108 * q^99

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−5.00000 3.00000 17.0000 0 −15.0000 −7.00000 −45.0000 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$+1$$
$$7$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.4.a.a 1
3.b odd 2 1 1575.4.a.l 1
5.b even 2 1 105.4.a.b 1
5.c odd 4 2 525.4.d.a 2
15.d odd 2 1 315.4.a.a 1
20.d odd 2 1 1680.4.a.u 1
35.c odd 2 1 735.4.a.j 1
105.g even 2 1 2205.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.b 1 5.b even 2 1
315.4.a.a 1 15.d odd 2 1
525.4.a.a 1 1.a even 1 1 trivial
525.4.d.a 2 5.c odd 4 2
735.4.a.j 1 35.c odd 2 1
1575.4.a.l 1 3.b odd 2 1
1680.4.a.u 1 20.d odd 2 1
2205.4.a.b 1 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(525))$$:

 $$T_{2} + 5$$ T2 + 5 $$T_{11} - 12$$ T11 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 5$$
$3$ $$T - 3$$
$5$ $$T$$
$7$ $$T + 7$$
$11$ $$T - 12$$
$13$ $$T + 30$$
$17$ $$T - 134$$
$19$ $$T + 92$$
$23$ $$T + 112$$
$29$ $$T + 58$$
$31$ $$T + 224$$
$37$ $$T - 146$$
$41$ $$T - 18$$
$43$ $$T + 340$$
$47$ $$T + 208$$
$53$ $$T - 754$$
$59$ $$T - 380$$
$61$ $$T - 718$$
$67$ $$T + 412$$
$71$ $$T + 960$$
$73$ $$T + 1066$$
$79$ $$T - 896$$
$83$ $$T + 436$$
$89$ $$T + 1038$$
$97$ $$T - 702$$